11.15 Specific Heats

The specific heat of a substance describes its absorption of heat in terms of its temperature change. In particular, the specific heat at constant volume, $C_v$, of a substance is the thermal energy that gets stored internally in the substance per unit temperature rise and per unit amount of substance.

As a first example, consider simple monatomic ideal gases, and in particular noble gases. Basic physics, or section 11.14.4, shows that for an ideal gas, the molecules have $\frac12{k_{\rm B}}T$ of translational kinetic energy in each of the three directions of a Cartesian coordinate system, where $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.38 10$\POW9,{-23}$ J/K is Boltzmann’s constant. So the specific heat per molecule is $\frac32{k_{\rm B}}$. For a kmol (6.02 10$\POW9,{26}$) of molecules instead of one, $k_{\rm B}$ becomes the “universal gas constant” $R_{\rm {u}}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 8.31 kJ/kmol K. Hence for a

\begin{displaymath}
\mbox{monatomic ideal gas: }
\bar{C}_v = {\textstyle\frac{3}{2}} R_{\rm {u}} = 12.5\mbox{ kJ/kmol K}
\end{displaymath} (11.63)

on a kmol basis. As figure 11.15 shows, this is very accurate for noble gases, including helium. (To get the more usual specific heat $C_v$ per kilogram instead of kmol, divide by the molar mass $M$. For example, for helium with two protons and two neutrons in its nucleus, the molar mass is about 4 kg/kmol, so divide by 4. In thermo books, you will probably find the molar mass values you need mislisted as molecular mass, without units. Just use the values and ignore the name and the missing units of kg/kmol. See the notations for more.)

Many important ideal gases, such as hydrogen, as well as the oxygen and nitrogen that make up air, are diatomic. Now a diatomic molecule can also have significant kinetic energy due to rotation of the two atoms around each other. Note that mathematically, the entire atom can rotate around two Cartesian axes orthogonal to the axis connecting the two nuclei, each motion contributing to the kinetic energy. Classical physics, in particular the “equipartition theorem,” would then predict that each of these rotational motions has $\frac12{k_{\rm B}}$ of kinetic energy too, raising the specific heat to $\frac52{k_{\rm B}}$. Well, figure 11.15 shows that at room temperature this is quite accurate for common diatomic gases. But note that these experimental data show that there are problems, both at very low temperatures, and at very high ones.

And there are major theoretical problems too. How about rotation around the axis through the nuclei? The nuclei are not point masses but have a small size. And the electrons, which are much further from the axis through the nuclei have very small mass compared to the nuclei, but it is not zero. So classical physics would demand that there is an additional $\frac12{k_{\rm B}}$ of kinetic energy in that rotation. It would require very fast rotation to get $\frac12{k_{\rm B}}$, so maybe there is some explanation in there somewhere. But another thing could surely not be ignored based on classical physics: even if you model the molecule as two point masses, (the nuclei), connected to each other, their connection is surely not rigid. The molecules should vibrate relative to each other. There should be $\frac12{k_{\rm B}}$ kinetic energy of vibration and $\frac12{k_{\rm B}}$ potential energy of vibration, assuming a linear force / displacement relationship. So even assuming a theoretical model of two connected point masses, the specific heat should be $\frac72{k_{\rm B}}$. It was a big problem for classical physics. (Not to mention that, as Maxwell noted, if you really take classical theory at face value, things get much worse still, since the individual internal part of the atoms, like the individual electrons and quarks in the nuclei, would each have to absorb their own thermal energy too. This would produce an enormously high specific heat.)

Figure 11.15: Specific heat at constant volume of gases. Temperatures from absolute zero to 1,200 K. Data from NIST-JANAF and AIP.
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...t(15,57){\makebox(0,0){He, Ne, Ar, Kr, \ldots}}
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Hydrogen in particular was a mystery before the advent of quantum mechanics: at low temperatures it would behave as a monatomic gas, with a specific heat of $\frac32{k_{\rm B}}$ per molecule, figure 11.15. That meant that the molecule had to be translating only, like a monatomic gas. How could the random thermal motion not cause any angular rotation of the two atoms around their mutual center of gravity, nor vibration of the atoms towards and away from each other?

Quantum mechanics solved this problem. In quantum mechanics the angular momentum of the molecule, as well as the harmonic oscillation energy, are quantized. For hydrogen at low temperatures, the typical available thermal energy $\frac12{k_{\rm B}}T$ is not enough to reach the next level for either. No energy can therefore be put into rotation of the molecule, nor in increased internal vibration. So hydrogen does indeed have the specific heat of monatomic gases at low temperatures, weird as it may seem. The rotational and vibrational motion are frozen out.

At normal temperatures, there is enough thermal energy to reach nonzero angular momentum states, but not higher vibrational ones, and the specific heat becomes

\begin{displaymath}
\mbox{typical diatomic ideal gas: }
\bar{C}_v = {\textstyle\frac{5}{2}} R_{\rm {u}} = 20.8 \mbox{ kJ/kmol K}.
\end{displaymath} (11.64)

Actual values for hydrogen, nitrogen and oxygen at room temperature are 2.47, 2.50, and 2.53 $R_{\rm {u}}$.

For high enough temperature, the vibrational modes will start becoming active, and the specific heats will start inching up towards 3.5 $R_{\rm {u}}$ (and beyond), figure 11.15. But it takes to temperatures of 1,000 K (hydrogen), 600 K (nitrogen), or 400 K (oxygen) before there is a 5% deviation from the 2.5 $R_{\rm {u}}$ value.

These differences may be understood from the solution of the harmonic oscillator derived in chapter 4.1. The energy levels of an harmonic oscillator are apart by an amount $\hbar\omega$, where $\omega$ is the angular frequency. Modeled as a simple spring-mass system, $\omega$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\sqrt{c/m}$, where $c$ is the equivalent spring stiffness and $m$ the equivalent mass. So light atoms that are bound tightly will require a lot of energy to reach the second vibrational state. Hydrogen is much lighter than nitrogen or oxygen, explaining the higher temperature before vibration become important for it. The molar masses of nitrogen and oxygen are similar, but nitrogen is bound with a triple bond, and oxygen only a double one. So nitrogen has the higher stiffness of the two and vibrates less readily.

Following this reasoning, you would expect fluorine, which is held together with only a single covalent bond, to have a higher specific heat still, and figure 11.15 confirms it. And chlorine and bromine, also held together by a single covalent bond, but heavier than fluorine, approach the classical value 3.5 $R_{\rm {u}}$ fairly closely at normal temperatures: Cl$_2$ has 3.08 $R_{\rm {u}}$ and Br$_2$ 3.34 $R_{\rm {u}}$.

For solids, the basic classical idea in terms of atomic motion would be that there would be $\frac32R_{\rm {u}}$ per atom in kinetic energy and $\frac32R_{\rm {u}}$ in potential energy:

\begin{displaymath}
\mbox{law of Dulong and Petit: }
\bar{C}_v = 3 R_{\rm {u}} = 25 \mbox{ kJ/kmol K}.
\end{displaymath} (11.65)

Not only is this a nice round number, it actually works well for a lot of relatively simple solids at room temperature. For example, aluminum is 2.91 $R_{\rm {u}}$, copper 2.94, gold 3.05, iron 3.02.

Note that typically for solids $\bar{C}_p$, the heat added per unit temperature change at constant pressure is given instead of $\bar{C}_v$. However, unlike for gases, the difference between $\bar{C}_p$ and $\bar{C}_v$ is small for solids and most liquids and will be ignored here.

Dulong and Petit also works for liquid water if you take it per kmol of atoms, rather than kmol of molecules, but not for ice. Ice has 4.6 $R_{\rm {u}}$ per kmol of molecules and 1.5 $R_{\rm {u}}$ per kmol of atoms. For molecules, certainly there is an obvious problem in deciding how many pieces you need to count as independently moving units. A value of 900 $R_{\rm {u}}$ for paraffin wax (per molecule) found at Wikipedia may sound astonishing, until you find elsewhere at Wikipedia that its chemical formula is C$_{25}$H$_{52}$. It is still quite capable of storing a lot of heat per unit weight too, in any case, but nowhere close to hydrogen. Putting $\frac52{k_{\rm B}}T$ in a molecule with the tiny molecular mass of just about two protons is the real way to get a high heat content per unit mass.

Complex molecules may be an understandable problem for the law of Dulong and Petit, but how come that diamond has about 0.73 $R_{\rm {u}}$, and graphite 1.02 $R_{\rm {u}}$, instead of 3 as it should? No molecules are involved there. The values of boron at 1.33 $R_{\rm {u}}$ and beryllium at 1.98 $R_{\rm {u}}$ are much too low too, though not as bad as diamond or graphite.

Figure 11.16: Specific heat at constant pressure of solids. Temperatures from absolute zero to 1,200 K. Carbon is diamond; graphite is similar. Water is ice and liquid. Data from NIST-JANAF, CRC, AIP, Rohsenow et al.
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Actually, it turns out, figure 11.16, that at much higher temperatures diamond does agree nicely with the Dulong and Petit value. Conversely, if the elements that agree well with Dulong and Petit at room temperature are cooled to low temperatures, they too have a specific heat that is much lower than the Dulong and Petit value. For example, at 77 K, aluminum has 1.09 $R_{\rm {u}}$, copper 1.5, and diamond 0.01.

It turns out that for all of them a characteristic temperature can by found above which the specific heat is about the Dulong and Petit value, but below which the specific heat starts dropping precariously. This characteristic temperature is called the Debye temperature. For example, aluminum, copper, gold, and iron have Debye temperatures of 394, 315, 170, and 460 K, all near or below room temperature, and their room temperature specific heats agree reasonably with the Dulong and Petit value. Conversely, diamond, boron, and beryllium have Debye temperatures of 1,860, 1,250, and 1,000 K, and their specific heats are much too low at room temperature.

The lack of heat capacity below the Debye temperature is again a matter of frozen out vibrational modes, like the freezing out of the vibrational modes that gave common diatomic ideal gases a heat capacity of only $\frac52R_{\rm {u}}$ instead of $\frac72R_{\rm {u}}$. Note for example that carbon, boron and beryllium are light atoms, and that the diamond structure is particularly stiff, just the properties that froze out the vibrational modes in diatomic gas molecules too. However, the actual description is more complex than for a gas: if all vibrations were frozen out in a solid, there would be nothing left.

Atoms in a solid cannot be considered independent harmonic oscillators like the pairs of atoms in diatomic molecules. If an atom in a solid moves, its neighbors are affected. The proper way to describe the motion of the atoms is in terms of crystal-wide vibrations, such as those that in normal continuum mechanics describe acoustical waves. There are three variants of such waves, corresponding to the three independent directions the motion of the atoms can take with respect to the propagation direction of the wave. The atoms can move in the same direction, like in the acoustics of air in a pipe, or in a direction normal to it, like surface waves in water. Those are called longitudinal and transverse waves respectively. If there is more than one atom in the basis from which the solid crystal is formed, the atoms in a basis can also vibrate relative to each other’s position in high-frequency vibrations called optical modes. However, after such details are accounted for, the classical internal energy of a solid is still the Dulong and Petit value.

Enter quantum mechanics. Just like quantum mechanics says that the energy of vibrating electromagnetic fields of frequency $\omega$ comes in discrete units called photons, with energy $\hbar\omega$, it says that the energy of crystal vibrations comes in discrete units called phonons with energy $\hbar\omega$. As long as the typical amount of heat energy, ${k_{\rm B}}T$, is larger than the largest of such phonon energies, the fact that the energy levels are discrete make no real difference, and classical analysis works fine. But for lower temperatures, there is not enough energy to create the high-energy phonons and the specific heat will be less. The representative temperature $T_D$ at which the heat energy ${k_{\rm B}}T_D$ becomes equal to the highest phonon energies $\hbar\omega$ is the Debye temperature. (The Debye analysis is not exact except for low energies, and the definitions of Debye temperature vary somewhat. See section 11.14.6 for more details.)

Quantum mechanics did not just solve the low temperature problems for heat capacity; it also solved the electron problem. That problem was that classically electrons in at least metals too should have $\frac32{k_{\rm B}}T$ of kinetic energy, since electrical conduction meant that they moved independently of the atoms. But observations showed it was simply not there. The quantum mechanical explanation was the Fermi-Dirac distribution of figure 6.11: only a small fraction of the electrons have free energy states above them within a distance of order ${k_{\rm B}}T$, and only these can take on heat energy. Since so few electrons are involved, the amount of energy they absorb is negligible except at very low temperatures. At very low temperatures, the energy in the phonons becomes very small, and the conduction electrons in metals then do make a difference.

Also, when the heat capacity due to the atom vibrations levels off to the Dulong and Petit value, that of the valence electrons keeps growing. Furthermore, at higher temperatures the increased vibrations lead to increased deviations in potential from the harmonic oscillator relationship. Wikipedia, Debye model, says anharmonicity causes the heat capacity to rise further; apparently authoritative other sources say that it can either increase or decrease the heat capacity. In any case, typical solids do show an increase of the heat capacity above the Dulong and Petit value at higher temperatures, figure 11.16.